Why is the order in dictionaries and sets arbitrary?
The other answers to this question are excellent and well written. The OP asks "how" which I interpret as "how do they get away with" or "why".
The Python documentation says dictionaries are not ordered because the Python dictionary implements the abstract data type associative array. As they say
the order in which the bindings are returned may be arbitrary
In other words, a computer science student cannot assume that an associative array is ordered. The same is true for sets in math
the order in which the elements of a set are listed is irrelevant
and computer science
a set is an abstract data type that can store certain values, without any particular order
Implementing a dictionary using a hash table is an implementation detail that is interesting in that it has the same properties as associative arrays as far as order is concerned.
This is more a response to Python 3.41 A set before it was closed as a duplicate.
The others are right: don't rely on the order. Don't even pretend there is one.
That said, there is one thing you can rely on:
list(myset) == list(myset)
That is, the order is stable.
Understanding why there is a perceived order requires understanding a few things:
That Python uses hash sets,
How CPython's hash set is stored in memory and
How numbers get hashed
From the top:
A hash set is a method of storing random data with really fast lookup times.
It has a backing array:
# A C array; items may be NULL,
# a pointer to an object, or a
# special dummy object
_ _ 4 _ _ 2 _ _ 6
We shall ignore the special dummy object, which exists only to make removes easier to deal with, because we won't be removing from these sets.
In order to have really fast lookup, you do some magic to calculate a hash from an object. The only rule is that two objects which are equal have the same hash. (But if two objects have the same hash they can be unequal.)
You then make in index by taking the modulus by the array length:
hash(4) % len(storage) = index 2
This makes it really fast to access elements.
Hashes are only most of the story, as hash(n) % len(storage) and hash(m) % len(storage) can result in the same number. In that case, several different strategies can try and resolve the conflict. CPython uses "linear probing" 9 times before doing complicated things, so it will look to the left of the slot for up to 9 places before looking elsewhere.
CPython's hash sets are stored like this:
A hash set can be no more than 2/3 full. If there are 20 elements and the backing array is 30 elements long, the backing store will resize to be larger. This is because you get collisions more often with small backing stores, and collisions slow everything down.
The backing store resizes in powers of 4, starting at 8, except for large sets (50k elements) which resize in powers of two: (8, 32, 128, ...).
So when you create an array the backing store is length 8. When it is 5 full and you add an element, it will briefly contain 6 elements. 6 > ²⁄₃·8 so this triggers a resize, and the backing store quadruples to size 32.
Finally, hash(n) just returns n for numbers (except -1 which is special).
So, let's look at the first one:
v_set = {88,11,1,33,21,3,7,55,37,8}
len(v_set) is 10, so the backing store is at least 15(+1) after all items have been added. The relevant power of 2 is 32. So the backing store is:
__ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __
We have
hash(88) % 32 = 24
hash(11) % 32 = 11
hash(1) % 32 = 1
hash(33) % 32 = 1
hash(21) % 32 = 21
hash(3) % 32 = 3
hash(7) % 32 = 7
hash(55) % 32 = 23
hash(37) % 32 = 5
hash(8) % 32 = 8
so these insert as:
__ 1 __ 3 __ 37 __ 7 8 __ __ 11 __ __ __ __ __ __ __ __ __ 21 __ 55 88 __ __ __ __ __ __ __
33 ← Can't also be where 1 is;
either 1 or 33 has to move
So we would expect an order like
{[1 or 33], 3, 37, 7, 8, 11, 21, 55, 88}
with the 1 or 33 that isn't at the start somewhere else. This will use linear probing, so we will either have:
↓
__ 1 33 3 __ 37 __ 7 8 __ __ 11 __ __ __ __ __ __ __ __ __ 21 __ 55 88 __ __ __ __ __ __ __
or
↓
__ 33 1 3 __ 37 __ 7 8 __ __ 11 __ __ __ __ __ __ __ __ __ 21 __ 55 88 __ __ __ __ __ __ __
You might expect the 33 to be the one that's displaced because the 1 was already there, but due to the resizing that happens as the set is being built, this isn't actually the case. Every time the set gets rebuilt, the items already added are effectively reordered.
Now you can see why
{7,5,11,1,4,13,55,12,2,3,6,20,9,10}
might be in order. There are 14 elements, so the backing store is at least 21+1, which means 32:
__ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __
1 to 13 hash in the first 13 slots. 20 goes in slot 20.
__ 1 2 3 4 5 6 7 8 9 10 11 12 13 __ __ __ __ __ __ 20 __ __ __ __ __ __ __ __ __ __ __
55 goes in slot hash(55) % 32 which is 23:
__ 1 2 3 4 5 6 7 8 9 10 11 12 13 __ __ __ __ __ __ 20 __ __ 55 __ __ __ __ __ __ __ __
If we chose 50 instead, we'd expect
__ 1 2 3 4 5 6 7 8 9 10 11 12 13 __ __ __ __ 50 __ 20 __ __ __ __ __ __ __ __ __ __ __
And lo and behold:
{1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 20, 50}
#>>> {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 50, 20}
pop is implemented quite simply by the looks of things: it traverses the list and pops the first one.
This is all implementation detail.
"Arbitrary" isn't the same thing as "non-determined".
What they're saying is that there are no useful properties of dictionary iteration order that are "in the public interface". There almost certainly are many properties of the iteration order that are fully determined by the code that currently implements dictionary iteration, but the authors aren't promising them to you as something you can use. This gives them more freedom to change these properties between Python versions (or even just in different operating conditions, or completely at random at runtime) without worrying that your program will break.
Thus if you write a program that depends on any property at all of dictionary order, then you are "breaking the contract" of using the dictionary type, and the Python developers are not promising that this will always work, even if it appears to work for now when you test it. It's basically the equivalent of relying on "undefined behaviour" in C.
Note: This answer was written before the implementation of the
dicttype changed, in Python 3.6. Most of the implementation details in this answer still apply, but the listing order of keys in dictionaries is no longer determined by hash values. The set implementation remains unchanged.
The order is not arbitrary, but depends on the insertion and deletion history of the dictionary or set, as well as on the specific Python implementation. For the remainder of this answer, for 'dictionary', you can also read 'set'; sets are implemented as dictionaries with just keys and no values.
Keys are hashed, and hash values are assigned to slots in a dynamic table (it can grow or shrink based on needs). And that mapping process can lead to collisions, meaning that a key will have to be slotted in a next slot based on what is already there.
Listing the contents loops over the slots, and so keys are listed in the order they currently reside in the table.
Take the keys 'foo' and 'bar', for example, and lets assume the table size is 8 slots. In Python 2.7, hash('foo') is -4177197833195190597, hash('bar') is 327024216814240868. Modulo 8, that means these two keys are slotted in slots 3 and 4 then:
>>> hash('foo')
-4177197833195190597
>>> hash('foo') % 8
3
>>> hash('bar')
327024216814240868
>>> hash('bar') % 8
4
This informs their listing order:
>>> {'bar': None, 'foo': None}
{'foo': None, 'bar': None}
All slots except 3 and 4 are empty, looping over the table first lists slot 3, then slot 4, so 'foo' is listed before 'bar'.
bar and baz, however, have hash values that are exactly 8 apart and thus map to the exact same slot, 4:
>>> hash('bar')
327024216814240868
>>> hash('baz')
327024216814240876
>>> hash('bar') % 8
4
>>> hash('baz') % 8
4
Their order now depends on which key was slotted first; the second key will have to be moved to a next slot:
>>> {'baz': None, 'bar': None}
{'bar': None, 'baz': None}
>>> {'bar': None, 'baz': None}
{'baz': None, 'bar': None}
The table order differs here, because one or the other key was slotted first.
The technical name for the underlying structure used by CPython (the most commonly used Python implemenation) is a hash table, one that uses open addressing. If you are curious, and understand C well enough, take a look at the C implementation for all the (well documented) details. You could also watch this Pycon 2010 presentation by Brandon Rhodes about how CPython dict works, or pick up a copy of Beautiful Code, which includes a chapter on the implementation written by Andrew Kuchling.
Note that as of Python 3.3, a random hash seed is used as well, making hash collisions unpredictable to prevent certain types of denial of service (where an attacker renders a Python server unresponsive by causing mass hash collisions). This means that the order of a given dictionary or set is then also dependent on the random hash seed for the current Python invocation.
Other implementations are free to use a different structure for dictionaries, as long as they satisfy the documented Python interface for them, but I believe that all implementations so far use a variation of the hash table.
CPython 3.6 introduces a new dict implementation that maintains insertion order, and is faster and more memory efficient to boot. Rather than keep a large sparse table where each row references the stored hash value, and the key and value objects, the new implementation adds a smaller hash array that only references indices in a separate 'dense' table (one that only contains as many rows as there are actual key-value pairs), and it is the dense table that happens to list the contained items in order. See the proposal to Python-Dev for more details. Note that in Python 3.6 this is considered an implementation detail, Python-the-language does not specify that other implementations have to retain order. This changed in Python 3.7, where this detail was elevated to be a language specification; for any implementation to be properly compatible with Python 3.7 or newer it must copy this order-preserving behaviour. And to be explicit: this change doesn't apply to sets, as sets already have a 'small' hash structure.
Python 2.7 and newer also provides an OrderedDict class, a subclass of dict that adds an additional data structure to record key order. At the price of some speed and extra memory, this class remembers in what order you inserted keys; listing keys, values or items will then do so in that order. It uses a doubly-linked list stored in an additional dictionary to keep the order up-to-date efficiently. See the post by Raymond Hettinger outlining the idea. OrderedDict objects have other advantages, such as being re-orderable.
If you wanted an ordered set, you can install the oset package; it works on Python 2.5 and up.